{"id":"n-86c6e99e3fb4b8b5","revision":0,"first_index":0,"fingerprint":"f6-d80136a6885fd53b","name":"reversal of basis_map_A013609 of Triangle read by rows: T(n,k) = binomial(k,n-k).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2","2"],["4","8","3"],["8","24","20","5"],["16","64","84","44","8"],["32","160","288","236","90","13"],["64","384","880","992","588","176","21"],["128","896","2496","3600","2928","1356","334","34"],["256","2048","6720","11840","12320","7808","2964","620","55"],["512","4608","17408","36288","46176","37392","19376","6228","1132","89"],["1024","10240","43776","105472","159040","157440","104080","45568","12696","2040","144"],["2048","22528","107520","294144","513536","602560","486560","271360","102792","25268","3638","233"],["4096","49152","259072","793600","1575936","2142208","2049600","1394880","672160","224288","49320","6432","377"],["8192","106496","614400","2083840","4641280","7181568","7960064","6415040","3768480","1597680","476256","94728","11290","610"],["16384","229376","1437696","5349376","13212672","22949888","28962048","27031552","18791360","9703040","3670928","988640","179492","19696","987"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","15","57","216","819","3105","11772","44631","169209","641520","2432187","9221121","34959924","132543135"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["2"],["2"]]},"terms":[{"coeff":{"coeffs":[["4","0","-5"],["4","8","5"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-4","-12","-5"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":175271,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-b509547f79e5aeee"]},{"id":175272,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-b509547f79e5aeee"]},{"id":175273,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-b509547f79e5aeee"]}],"region_exact":{},"incoming_edges":[{"id":"e-b509547f79e5aeee","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-86c6e99e3fb4b8b5","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"2ebfd5ebab14bdd600dd7c5848e1cd3da7a87637c7dafba9d0f86f2f4e9099c7","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-0836bf76d9ba1ba5","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(n,k) = binomial(k,n-k).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-bc26814fe7510caa","kind":"fingerprint_related","node_a_id":"n-0836bf76d9ba1ba5","node_b_id":"n-86c6e99e3fb4b8b5","fingerprint":"f6-d80136a6885fd53b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d80136a6885fd53b"},"other_id":"n-0836bf76d9ba1ba5","other":{"id":"n-0836bf76d9ba1ba5","name":"basis_map_A013609 of Triangle read by rows: T(n,k) = binomial(k,n-k).","first_index":0,"fingerprint":"f6-d80136a6885fd53b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","2"],["3","8","4"]],"card_rows":[["1"],["2","2"],["3","8","4"],["5","20","24","8"]],"row_sums_preview":["1","4","15","57","216","819","3105","11772"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-21854f4257effb01","steps":["basis A013609"],"edges":["e-38aa524f6e49a0d8"],"text":"A030528 \u2192 basis A013609"},"headline":"A030528 \u2192 basis A013609","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:   1;\n  1:   2,    2;\n  2:   4,    8,     3;\n  3:   8,   24,    20,     5;\n  4:  16,   64,    84,    44,     8;\n  5:  32,  160,   288,   236,    90,    13;\n  6:  64,  384,   880,   992,   588,   176,    21;\n  7: 128,  896,  2496,  3600,  2928,  1356,   334,   34;\n  8: 256, 2048,  6720, 11840, 12320,  7808,  2964,  620,   55;\n  9: 512, 4608, 17408, 36288, 46176, 37392, 19376, 6228, 1132, 89;\n","recurrence_sentence":"The row polynomials satisfy (2*n+2)*P(n,x) = (4*n+4+8*n*x+(5*n-5)*x^2)*P(n-1,x) - (4*x+12*x^2+5*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n + 4 + 8*n*x + (5*n - 5)*x^2)*P[n - 1, x] - (4*x + 12*x^2 + 5*x^3)*D[P[n - 1, x], x])/(2*n + 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4*n + 4 + 8*n*x + (5*n - 5)*x^2)*v[i - 1] - (4*x + 12*x^2 + 5*x^3)*deriv(v[i - 1], x))/(2*n + 2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-21854f4257effb01","steps":["basis A013609","reverse"],"edges":["e-38aa524f6e49a0d8","e-b509547f79e5aeee"],"text":"A030528 \u2192 basis A013609 \u2192 reverse"},"headline":"A030528 \u2192 basis A013609 \u2192 reverse","chain":null}
